Island-size distribution and capture numbers in three-dimensional ...

PHYSICAL REVIEW B 71, 245411 共2005兲

Island-size distribution and capture numbers in three-dimensional nucleation: Comparison with mean-field behavior Feng Shi,* Yunsic Shim,† and Jacques G. Amar‡ Department of Physics & Astronomy, University of Toledo, Toledo, Ohio 43606, USA 共Received 4 February 2005; published 21 June 2005兲 The scaling of the monomer and island densities, island-size distribution 共ISD兲, capture-number distribution 共CND兲, and capture-zone distribution is studied as a function of the fraction of occupied sites 共coverage兲 and ratio D / F of the monomer hopping rate D to the 共per site兲 monomer creation rate F in a three-dimensional 共3D兲 point-island model of irreversible nucleation and island growth. Our model is a 3D analog of submonolayer growth and may also be viewed as a simplified model of the early stages of vacancy cluster nucleation and growth under irradiation. Good agreement is found between mean-field 共MF兲 rate-equation results for the average island and monomer densities and our simulation results. In addition, due to the decreased influence of correlations and fluctuations in 3D, the scaled CND depends only weakly on the island-size. As a result, the scaled ISD is significantly sharper than obtained in 2D and diverges with increasing D / F. However, for large D / F both the scaled ISD and the scaled CND differ from the MF prediction. In particular, the scaled ISD diverges more slowly than the MF prediction while the asymptotic divergence occurs at a value of the scaled island size which is larger than the MF prediction. These results are further supported by an analysis of the asymptotic scaled capture-number distribution. DOI: 10.1103/PhysRevB.71.245411

PACS number共s兲: 81.15.Aa, 68.55.Ac, 68.43.Jk

I. INTRODUCTION

Cluster nucleation and growth is the central feature of many physical processes. The nucleation and growth of islands in submonolayer epitaxial growth has been studied intensively both experimentally1–10 and theoretically11–30 since the structures formed in the submonolayer regime can strongly influence the morphology and properties of the resultant multilayer film. For example, recently considerable theoretical effort has been carried out towards an understanding of the scaling properties of the island-size distribution Ns共␪兲 共where Ns is the number of islands of size s at coverage ␪兲 in submonolayer growth.11–30 In the precoalescence regime the island-size distribution satisfies the scaling form14,15 N s共 ␪ 兲 =

冉冊

␪ s f , S2 S

共1兲

where S is the average island size and the scaling function f共u兲 depends on the critical island size and island morphology.18 One of the standard tools used in these studies is the rateequation 共RE兲 approach.11,12,31 In this approach the coverage dependence of the island-size distribution 共ISD兲 is calculated through a set of deterministic reaction-diffusion equations which involve a set of rate coefficients usually called capture numbers.11,12 For the irreversible growth of point islands, rate equations valid in the precoalescence regime may be written in the form ⬁

⬁

dN1 = 1 − 2R␴1N21 − RN1 ␴sNs − ␬1N1 − ␬sNs , 共2兲 d␪ s=2 s=1

兺

1098-0121/2005/71共24兲/245411共6兲/$23.00

兺

dNs = R␴s−1N1Ns−1 − R␴sN1Ns + ␬s−1Ns−1 − ␬sNs , d␪

共3兲

where the capture numbers ␴s 共␴1兲 correspond to the average capture rate of diffusing monomers by islands of size s 共monomers兲, R = D / F is the ratio of the monomer diffusion rate to the deposition rate, and the terms with ␬s 共␬s = 1 for point islands兲 correspond to direct impingement. We note that the central problem in the RE approach is the determination of the size- and coverage-dependent capture numbers ␴s共␪兲. The simplest possible assumption for the capture-number distribution 共CND兲 is the mean-field assumption ␴s共␪兲 = ␴av. While such an assumption may be adequate to describe the scaling of the average island density N and monomer density N1 with coverage and D / F, it is not adequate to describe the ISD. For example, using two-dimensional 共2D兲 kinetic Monte Carlo 共KMC兲 simulations, Bartelt and Evans19 showed that even for point islands there is a nontrivial dependence of the capture number on the island size due to the correlation between the island size and capture zone. In addition, they showed that in the asymptotic limit of large D / F, the scaled ISD is related to the scaled CND as

冋冕

u

f共u兲 = f共0兲exp

0

dx

册

2z − 1 − C⬘共x兲 , C共x兲 − zx

共4兲

where C共s / S兲 = ␴s / ␴av is the scaled CND, z is the dynamical exponent describing the dependence of the average island size on coverage 共S ⬃ ␪ z兲, and f共0兲 is determined by the normalization condition 兰⬁0 du f共u兲 = 1. We note that for irreversible growth of point islands as is considered here, one has z = 2 / 3. As can be seen from Eq. 共4兲, if C共u兲 ⬎ zu, then no divergence will occur. However, if C共u兲 crosses zu at

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some value uc, then a divergence in the asymptotic ISD will occur if C⬘共uc兲 ⬍ 2z − 1. This result highlights the importance of understanding the capture-number distribution in order to understand the scaling behavior of the ISD. For example, using this expression Bartelt and Evans were able to show19 that the usual meanfield 共MF兲 assumption that the capture number is independent of island size leads to a divergent ISD. On the other hand, by measuring the CND for various models in twodimensional submonolayer island growth, they showed that due to correlations, the actual CND is not mean field but depends strongly on the island size. Accordingly, the scaled ISD for 2D nucleation and growth does not diverge in the asymptotic limit. While the scaling behavior of the ISD and CND is now well understood for the case of two-dimensional nucleation and growth, the corresponding behavior has not been studied in three dimensions. This is of interest from a theoretical point of view, since we would like to understand to what extent fluctuations play a role in determining the scaled ISD and CND, and a comparison with 3D simulations would be quite useful in providing such understanding. In addition, the scaling behavior of the ISD in simple models of 3D nucleation and growth may also be important in understanding a variety of important processes. For example, the nucleation and growth of islands in bulk materials has attracted tremendous interest in recent years, as these processes create nanoparticles which as quantum dots are promising in fabricating light emission devices.32–34 Here we present results for the scaled island-size distribution and capture-number distribution obtained from KMC simulations of a simple point-island model of 3D nucleation and growth. For completeness, we also present the results of a self-consistent RE calculation which leads to good agreement with KMC simulations for the coverage dependence of the average island density N共␪兲 and monomer density N1共␪兲. We find that, due to the decreased role of fluctuations and correlations in three dimensions, the scaled ISD in 3D is significantly sharper than in 2D and appears to diverge with increasing D / F, while the asymptotic CND depends only weakly on island size. However, the asymptotic scaled ISD and CND still appear to deviate from the MF prediction. We attribute this deviation to the existence of geometric effects and correlations which, although reduced in 3D, still appear to play a role in three dimensions. This paper is organized as follows. In Sec. II we first describe our simulations and point-island model. In Sec. III we describe a self-consistent rate-equation approach to the calculation of the capture numbers. In Sec. IV we first present a comparison between our self-consistent RE results and KMC results for the average island and monomer densities. We then present our KMC results for the ISD and CND along with a comparison with the corresponding RE results. Results for the scaled capture-zone distribution 共CZD兲 are also presented. Finally, we discuss and summarize our results in Sec. V.

point-island model of irreversible nucleation and growth on a cubic lattice. Our model is a straightforward analog of the corresponding point-island model previously studied in two dimensions.19 However, it may also be considered to be a simple model of vacancy cluster nucleation and growth in solids. We note that a more realistic model would take into account the increase of the lateral dimension of an island with island size. However, in the asymptotic limit of large D / F the point-island approximation is appropriate for extended islands up to a finite coverage 共␪ 艋 0.01兲 since over this coverage range the average island separation is still significantly larger than the average island radius.26 In our model, monomers are randomly created throughout the lattice with creation rate F per site per unit time and then hop randomly in each of the six nearest-neighbor directions with hopping rate Dh. If a monomer lands on a site already occupied by another monomer or is created at such a site, then a dimer island is nucleated. Similarly, if a monomer lands on or is created at a site already occupied by an island, then that monomer is captured by that island and the island size increases by 1. As for 2D nucleation and growth the key parameter in this model is the ratio Rh = Dh / F of the monomer hopping rate to the 共per site兲 monomer creation rate or, equivalently, the ratio R = D / F = Rh / 6. In order to study the asymptotic scaling behavior, we have carried out simulations over a range of values of Rh ranging from 105 to 1010. To avoid finite-size effects, simulations were carried out over a range of different values of the system size L ranging from L = 160 to L = 450. In addition, our results were typically averaged over 200 runs to obtain good statistics. For each set of parameters the scaled ISD, CND, and CZD were obtained for coverages up to ␪ = 0.4, while the average island density N共␪兲 and monomer density N1共␪兲 were also measured. We note that in order to measure the capture-number distribution, the method outlined in Ref. 19 was used. In particular, the capture number ␴s共␪兲 was calculated using the expression ␴s共␪兲 = nsc / 共R⌬␪N1NsL3兲 where nsc is the number of monomer capture events corresponding to an island of size s during a very small coverage interval 共⌬␪ ⯝ 0.001兲. As in Ref. 19 the island size s at the beginning of the coverage interval was used when incrementing the counter nsc in order to obtain good statistics. We also note that in our capture-zone distribution calculations, the capture zone of an island was defined as corresponding to all monomer sites or empty sites which are closer to that island than any other island. If such a site was equally close to several islands, then that site’s contribution to the capture zone was equally distributed between the islands.

III. SELF-CONSISTENT RATE-EQUATION CALCULATION

As in Ref. 17 we consider a quasistatic diffusion equation for the monomer density n1共r , ␪ , ␾兲 surrounding an island of size s of the form

II. MODEL AND SIMULATIONS

ⵜ2n1共r, ␪, ␾兲 − ␰−2共n1 − N1兲 = 0,

In order to study the scaling behavior of the ISD and CND in 3D nucleation and growth, we have studied a simple

where N1 is the average monomer density and ␰ corresponds to an overall average capture term. For consistency with the

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共5兲

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ISLAND-SIZE DISTRIBUTION AND CAPTURE… ⬁ RE’s 共2兲 and 共3兲 we require ␰−2 = 2␴1N1 + 兺s=2 ␴sNs. We note that if we assume that ␴s = ␴1 = ␴, then this self-consistency condition may be written more simply as

␰−2 = ␴共N + 2N1兲.

共6兲

Assuming spherical symmetry Eq. 共5兲 may be written

冉 冊

˜1 1 d 2 dn r − ␰−2˜n1共r兲 = 0, r2 dr dr

共7兲

where ˜n1共r兲 = n1共r兲 − N1. The general solution is given by ˜n1共r兲 = A

cosh共r/␰兲 sinh共r/␰兲 +B . r r

共8兲

Using the boundary condition n1共Rs兲 = 0 共where Rs is the island radius兲 corresponding to irreversible growth, along with the asymptotic boundary condition n1共⬁兲 = N1, we obtain n1共r兲 = N1关1 − 共Rs/r兲e−共r−Rs兲/␰兴.

共9兲

Equating the microscopic flux of atoms near the island 4␲Rs2D关⳵n1 / ⳵r兴r=Rs to the corresponding macroscopic RElike term DN1␴s, we obtain, for the capture number,

␴s =

冉 冊

4␲Rs2 ⳵n1 N1 ⳵r

= 4␲Rs共1 + Rs/␰兲.

共10兲

r=Rs

We note that this result agrees with that of Talbot and Willis35 who carried out an analysis of the mean “sink” strength of voids in a random array of voids in an irradiated material. For the point-island model the island radius Rs is independent of island size 共i.e., Rs = R0兲 which implies

␴s = ␴ = 4␲R0共1 + R0/␰兲,

共11兲

where R0 is a model-dependent constant of order 1. Substituting the self-consistency condition 共6兲 for ␰ leads to the result

␴ ⯝ 4␲R0关1 + ␹共1 + 冑1 + 2/␹兲兴, ␹ = 共N + 2N1兲2␲R30

共12兲

FIG. 1. Comparison between KMC results 共symbols兲 and the corresponding RE results 共solid lines兲 for the monomer density N1 and island density N as a function of coverage for Dh / F = 105 共circles兲, 107 共triangles兲, and 109 共diamonds兲.

= 105 , 107, and 109 are shown, while the value of R0 共R0 = 1 / 3兲 was chosen to give the best fit to the KMC data. As can be seen, there is excellent agreement between the RE and KMC results over all coverages and for all values of Dh / F. Thus, as was previously found in two dimensions,17 the selfconsistent RE approach can be used to accurately predict average quantities such as the monomer and island density in three dimensions. We now consider the scaled ISD. Figure 2 shows the corresponding results for the scaled ISD obtained from KMC simulations 共symbols兲 at coverage ␪ = 0.2. As can be seen, the peak of the scaled ISD increases with increasing D / F while the island-size distribution becomes sharper, thus indicating a divergence in the asymptotic limit of infinite D / F. Also shown in Fig. 2 are the corresponding self-consistent RE results 共solid curves兲. The asymptotic MF result12,18 f共u兲 = 31 共1 − 2u / 3兲−1/2 correspond-

⬁ N = 兺s=2 Ns

and is the average island where density. Using this result the contracted rate equations for the monomer and island densities may be written dN1 = 1 − 2N1 − N − 2共D/F兲␴N21 − 共D/F兲␴N1N, d␪

共13兲

dN = N1 + 共D/F兲␴N21 . d␪

共14兲

We note that for large D / F, Eq. 共12兲 implies a capture number ␴ ⯝ 4␲R0 which does not depend on coverage or D / F. IV. RESULTS

Figure 1 shows a comparison between our KMC simulation results for the average monomer and island densities and the corresponding RE results obtained by numerically integrating Eqs. 共12兲–共14兲. The KMC results for Dh / F

FIG. 2. Scaled island-size distributions for Dh / F = 105, 107, and 109. KMC simulation results 共symbols兲, RE results 共solid lines兲, and asymptotic MF limit 共dashed curve兲.

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FIG. 3. Log-log plot of peak value of scaled ISD as function of Dh / F.

ing to infinite D / F is also shown 共dashed curve兲. As expected, the self-consistent MF RE results for the ISD approach the asymptotic MF prediction with increasing D / F. However, there are significant differences between the KMC results and the RE results. In particular, while there is good agreement between the KMC and RE results for small D / F, for large D / F the KMC results for the scaled ISD are significantly lower than the RE predictions. These differences are more dramatically indicated in Fig. 3 which shows the peak values of the scaled ISD obtained from both KMC simulations and RE calculations as a function of D / F. As can be seen, in both cases the peak value f pk共D / F兲 of the scaled ISD increases as a power law—i.e., f pk ⬃ 共D / F兲␾—thus indicating a divergent ISD in the asymptotic limit. However, the value of ␾ obtained from the KMC simulations 共␾ ⯝ 0.06兲 is significantly smaller than the value 共␾ ⯝ 0.08兲 obtained from our RE calculations. In order to understand these differences, we have also measured the scaled capture-number distribution C共s / S兲 in our KMC simulations for Dh / F = 105 – 1010 as shown in Fig. 4. The MF prediction corresponding to the horizontal dashed line C共u兲 = 1 is also shown. As can be seen, there are significant deviations between the KMC results and the MF prediction. In particular, C共u兲 is less than 1 for u ⬍ 1.3 while it increases rapidly with u for u ⬎ 1.3. We note that the MF prediction corresponds to an asymptotic divergence in the scaled ISD at the point ucMF = 3 / 2 where the MF CND crosses the line 2u / 3. On the other hand, for large D / F the scaled CND curves obtained from the KMC simulation appear to “pivot” with increasing D / F around a fixed point at uc ⯝ 1.55 which is also the point at which they cross the line 2u / 3. As indicated by Eq. 共4兲, a divergence in the asymptotic ISD will only occur if C⬘共uc兲 ⬍ 2z − 1 = 1 / 3. For all values of D / F considered here, the slope C⬘共uc兲 of the scaled CND at the crossing point uc ⯝ 1.55 is lower than the critical value 2z − 1 = 1 / 3 required to avoid a divergence. Furthermore, for large D / F the slope is well described by the expression

FIG. 4. KMC simulation results for scaled CND ␴s / ␴av versus scaled island size for Dh / F = 105 – 1010. Horizontal dashed line corresponds to MF CND; solid line corresponds to 共2 / 3兲共s / S兲.

C⬘共uc兲 = 共1 / 3兲exp关−a / 共D / F兲0.2兴 where a = 23.6. Thus C⬘共uc兲 is less than the critical value of 1 / 3 for all finite D / F although it approaches the critical value asymptotically. These results are consistent with the observed divergence in the scaled ISD with increasing D / F. However, the asymptotic divergence appears to occur at a point, uc ⯝ 1.55, which is somewhat beyond the point ucMF = 3 / 2 at which the MF ISD diverges. In order to better understand the asymptotic behavior we have also studied the dependence of the peak position u pk of the scaled ISD obtained from our KMC simulations as a function of D / F. In order to extrapolate the asymptotic behavior, the peak position u pk共D / F兲 was fit to the form u pk共D / F兲 = u pk共⬁兲 + c共D / F兲−␥ while the value of ␥ was varied to find the best fit. A similar fit was used to extrapolate the MF RE results. Figure 5 shows the corresponding results for the KMC simulations 共solid circles, ␥ ⯝ 1 / 9兲 as well as for

FIG. 5. Plot of u pk共D / F兲 and ucr共D / F兲 as a function of 共Dh / F兲−␥ for Dh / F ranging from 105 to 1010. Lines are fits as described in text while values of ␥ correspond to best fits.

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ISLAND-SIZE DISTRIBUTION AND CAPTURE…

V. DISCUSSION

FIG. 6. KMC simulation results for scaled capture zone zs / Z versus scaled island size s / S for Dh / F = 105 – 109.

the MF RE results 共open circles, ␥ ⯝ 1 / 5兲. As expected, for the MF RE results we find u MF pk 共⬁兲 ⯝ 3 / 2. However, for the KMC simulation results we find u pk共⬁兲 ⯝ 1.55 which is in good agreement with our results for the scaled CND. We note that the value of ␥ 共␥ ⯝ 1 / 9兲 obtained in our fits to the KMC results may be partially explained by noting that 共D / F兲1/9 corresponds to the ratio of the island separation to the island size. However, we have no similar explanation for the value of ␥ 共␥ ⯝ 0.20兲 obtained in our fits to the MF RE calculations. We have carried out a similar analysis of the crossing point ucr corresponding to the position at which the tails of successive ISD curves 共corresponding to values of D / F varying by a factor of 10兲 cross. In this case a fit similar to that used for the peak position—i.e., ucr共D / F兲 = ucr共⬁兲 + c共D / F兲−␥—was used and again the value of ␥ was varied to find the best fit. The corresponding results are also shown in Fig. 5. We note that the best-fit value of ␥ 共␥ ⯝ 0.24兲 used for the KMC crossing points 共upper solid symbols兲 is significantly different from that used to fit the scaling of the KMC peak position 共␥ ⯝ 1 / 9兲 and is much closer to that used for the MF RE results 共upper open symbols兲. However, the asymptotic value of the crossing point ucr共⬁兲 ⯝ 1.55 is still in good agreement with the asymptotic value for the peak position. Similarly, for the MF RE results we again find MF 共⬁兲 ⯝ 3 / 2 and ␥ ⯝ 1 / 5. Thus these results confirm that, ucr as already indicated in Figs. 2–4, although the asymptotic scaled ISD diverges with increasing D / F, it is significantly different from the MF prediction. For completeness, we have also measured the scaled CZD as shown in Fig. 6. The shape of the scaled CZD is similar to that of the scaled CND. In particular, it is relatively “flat” for u ⬍ 1.3, while for u ⬎ 1.3 it increases rapidly with island size for large D / F. However, just as for the case of 2D submonolayer nucleation,19 the scaled CZD is quite different from the scaled CND. This difference is most likely due to the effects of geometry as well as fluctuations which imply that an adatom in the capture zone of a given island is not necessarily captured by that island.

In order to understand the role of fluctuations and geometry in irreversible nucleation and growth, and also to compare with results obtained in two dimensions, we have carried out simulations of a simple point-island model of irreversible island nucleation and growth in three dimensions. We have also presented a self-consistent RE calculation similar to that previously carried out by Bales and Chrzan17 in two dimensions. In contrast to the case of island nucleation and growth in 2D, we find that the peak of the scaled ISD increases and the distribution becomes sharper with increasing D / F, thus leading to a divergence in the asymptotic limit. However, while good agreement is found between the self-consistent RE results for the average island and monomer densities and our simulation results, there is poor agreement between the MF RE results for the scaled ISD and simulations. In particular, the scaled island-size distribution obtained in KMC simulations diverges more slowly than the MF prediction. By directly measuring the scaled capture-number distribution for different values of D / F we have found that, in contrast to the MF assumption, the scaled CND depends weakly on island size. In addition, our analysis of the dependence of the scaled CND on D / F clearly indicates that the asymptotic CND is also different from the MF prediction. In particular, the asymptotic CND appears to cross the line 2u / 3 at a value uc ⯝ 1.55 which is somewhat larger than the MF prediction uc = 3 / 2. As indicated by Eq. 共1兲, this leads to a divergence of the ISD at a value uc ⯝ 1.55 which is somewhat larger than the value 3 / 2 predicted by MF theory. These results are further supported by our analysis of the asymptotic behavior of the ISD peak position and crossing point which indicate that in both cases there is an asymptotic divergence at a scaled island size uc ⯝ 1.55 which is somewhat beyond the MF prediction. This “bending” of the CND away from the MF value C共u兲 = 1 for large u also leads to a decreased value of the numerator in Eq. 共1兲, thus explaining the “decrease” in the peak of the ISD compared to the MF prediction. Thus, we conclude that although the scaled ISD and CND in 3D are significantly closer to the MF prediction than in 2D, in the asymptotic limit of large D / F the scaled CND is still not completely independent of island size as predicted by MF theory. We believe that this is due to the effects of fluctuations and geometry, which still appear to play a significant role in 3D. We note that since we have been primarily interested in the asymptotic behavior of the point-island model, the results shown here have primarily focussed on the behavior at a relatively high coverage—i.e., ␪ = 0.2. While at higher coverages the differences between our KMC results and the MF prediction are even larger, we have also examined the behavior at significantly lower coverage ␪ ⯝ 0.05. In this case, we found that a similar discrepancy between the KMC results and the MF prediction also occurs, although it is significantly smaller for the same value of D / F. We conclude that at lower coverages, much higher values of D / F are needed to clearly see the asymptotic behavior. Finally, we note that for a more realistic model with 3D islands, the dependence of the capture number on island size is likely to be even stron-

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ger than for the point-island model studied here. Based on these results, we expect that the “critical dimension” dc for mean- field behavior for the point-island model is larger than 3 and is possibly equal to 4. In the future we plan to carry out parallel KMC simulations in four dimensions in order to see if this is the case.

*Electronic address: [email protected] †Electronic

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ACKNOWLEDGMENTS

This research was supported by the NSF through Grant No. DMR-0219328. We would also like to acknowledge grants of computer time from the Ohio Supercomputer Center 共Grant No. PJS0245兲.

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